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Limit problem from past exam.

  • Thread starter thepatient
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  • #1
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limit problem from past exam. :(

Homework Statement


Problem from an exam that I just did is really bugging me.

lim(x,y)->(0,0) (x*y2)/(x2+y4)






The attempt at a solution[/b]
At first I thought the limit didn't exist. I tried using paths, x=y1/2, but then I couldn't find a counter example for which a path approached at a different point. So then I tried using the squeeze theorem, and got that the limit is 0. :\

Was my approach to the problem correct? is the limit really 0?


[
 

Answers and Replies

  • #2
Dick
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You didn't say how you proved it. But yes, the limit is zero.
 
  • #3
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You didn't say how you proved it. But yes, the limit is zero.

Oh really? XD Well I assumed:

0<y2<x2+y4 (since x^2 and y^4 are always positive, and y2<y4

0<y2/x2+y4<1
Then multiplying x into the inequality:

0<x*y2/x2+y4<x

Then taking the limit:
lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y2/x2+y4<lim(x,y)->(0,0) x

Which gave me that:
0<lim(x,y)->(0,0) x*y2/x2+y4<0

which by the squeeze theorem, limit is zero. Was that correct...?
 
  • #4
Dick
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Oh really? XD Well I assumed:

0<y2<x2+y4

0<y2/x2+y4<1
Then multiplying x into the inequality:

0<x*y2/x2+y4<x

Then taking the limit:
lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y2/x2+y4<lim(x,y)->(0,0) x

Which gave me that:
0<lim(x,y)->(0,0) x*y2/x2+y4<0

which by the squeeze theorem, limit is zero. Was that correct...?
Well the starting point 0<y^2<x^2+y^4 is wrong. Suppose x=0 and y=(1/2). I would use polar coordinates.
 
  • #5
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Limit doesn't exist , try y=x^(1/2)
 
  • #6
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Aaah... I was starting it with polar coordinates at first and it seemed like the best way to go, but then it seemed to messy so I thought I was doing it wrong. I was running out of time and just left it like that. XD Thanks lots, at least maybe I have partial credit haha..
 
  • #7
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lim x->0 f(x,x^(1/2))=1/2
and lim x->0 f(x,0)=0
 
  • #8
Dick
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Limit doesn't exist , try y=x^(1/2)
Ooops. You are so right. Sorry.
 
  • #9
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Aww darn... it was a 3 part question worth 5 points total on that part of the exam, so I should have 10/3 credit hehe... The rest of the test was pretty good.
 

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